Technical note: Lognormal swap approximation in the Libor market model and its application

Koichi Matsumoto

Abstract

ABSTRACT

The market uses the Black formula to value both caps and swaptions. Because the Libor market model is one where the forward Libor rate has a lognormal volatility structure, it corresponds to the Black formula for a cap and is useful for valuing derivatives of the forward Libor rate. However, in implementing it we are faced with a serious problem. In the Libor market model a forward swap rate does not have a lognormal volatility structure, but it is assumed to have a lognormal volatility structure when its derivatives are evaluated and hedged. The Libor market model is not consistent with the convention of the swaption market. Therefore it is difficult to calibrate the volatility function in a way that is suitable for both the cap and the swaption market. In this paper we propose a useful approximation of a forward swap rate so that it has a lognormal volatility structure. In this case the Libor market model coincides with the Black formula for both caps and swaptions. Using this approximation we present a practical method for determining the implied correlation matrix uniquely and applying it. Further, we demonstrate a convexity adjustment by the implied correlation. Finally, we present some numerical results to demonstrate the usefulness of the approach.